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Q38E

Expert-verified
Found in: Page 338

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# if A is a ${\mathbf{2}}{\mathbf{×}}{\mathbf{2}}$matrix with t r A = 5 and det A = - 14 what are the eigenvalues of A?

The eigenvalues of A.

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],trA=5\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\lambda }^{2}-5\lambda -14=0\phantom{\rule{0ex}{0ex}}\left(\lambda +2\right)\left(\lambda -7\right)=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\lambda }_{1}=2,{\lambda }_{2}=7$

See the step by step solution

## Step 1: definition of matrix

A function is defined as a relationship between a set of inputs that each have one output.

$\mathrm{A}=\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right],\mathrm{tr}\mathrm{A}=5$

Now,

we solve: A = - 14

$\mathrm{det}\left(\mathrm{a}-\mathrm{\lambda l}\right)=0\phantom{\rule{0ex}{0ex}}\left|\begin{array}{cc}\mathrm{a}-\mathrm{\lambda }& \mathrm{b}\\ \mathrm{c}& \mathrm{d}-\mathrm{\lambda }=0\end{array}\right|$

## Step 2: definition of eigenvalues

The eigenvalue is a number that indicates how much variance exists in the data in that direction; in the example above, the eigenvalue is a number that indicates how spread out the data is on the line.

Multiply the matrix:

$\left(\mathrm{a}-\mathrm{\lambda }\right)\left(\mathrm{d}-\mathrm{\lambda }\right)-\mathrm{bc}=0\phantom{\rule{0ex}{0ex}}{\mathrm{\lambda }}^{2}-\left(\mathrm{a}+\mathrm{d}\right)\mathrm{\lambda }+\mathrm{ad}-\mathrm{bc}=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\mathrm{\lambda }}^{2}-5\mathrm{\lambda }-14=0\phantom{\rule{0ex}{0ex}}\left(1-\mathrm{\lambda }\right)\left(2-\mathrm{\lambda }\right)=0$

So,

${\lambda }_{1}=-2,{\lambda }_{2}=7$

Hence, ${\lambda }_{1}=-2,{\lambda }_{2}=7$